Problem 71 - Contemporary Precalculus

Question

A river flows from east to west. A swimmer on the south bank wants to swim to a point on the opposite shore directly north of her starting point. She can swim at \(2.8 \mathrm{mph},\) and there is a 1 -mph current in the river. In what direction should she head so as to travel directly north (that is, what angle should her path make with the south bank of the river)?

Step-by-Step Solution

Answer

Answer: The swimmer should swim at an angle of approximately 21.8 degrees relative to the south bank.

Step 1: Define the variables

Let's define the speed of the swimmer relative to water as \(v_S = 2.8 \mathrm{mph}\), the speed of current as \(v_C = 1 \mathrm{mph},\) and the angle between the swimmer's direction and south bank will be \(\theta.\)

Step 2: Represent swimmer's velocity as a vector

We can define the swimmer's velocity vector (relative to water) as \( \vec{v_S} = (v_{Sx}, v_{Sy})\), where \(v_{Sx}\) represents the east-west component, and \(v_{Sy}\) represents the north-south component. Using trigonometry, we have \(v_{Sx} = v_S \sin(\theta)\) \(v_{Sy} = v_S \cos(\theta)\)

Step 3: Represent the current's velocity as a vector

The current's velocity vector will be \(\vec{v_C} = (v_{Cx}, v_{Cy})\). Since the current is moving towards the west, we define its speed as moving towards the negative x-direction (west) and no motion in the north-south direction. Thus, \(v_{Cx} = -v_C = -1 \,\mathrm{mph}\) \(v_{Cy} = 0\)

Step 4: Find the net velocity vector

The swimmer's net velocity vector will be the vector sum of the swimmer's velocity and the current's velocity: \(\vec{v_N} = \vec{v_S} + \vec{v_C}\). Adding the respective components, we have: \(v_{Nx} = v_{Sx} + v_{Cx} = v_S \sin(\theta) - 1\) \(v_{Ny} = v_{Sy} + v_{Cy} = v_S \cos(\theta)\)

Step 5: Equate the net velocity's components to find the angle \(θ\)

Since the swimmer is going directly north, there is no east-west velocity component. So, \(v_{Nx} = 0\). Equating \(v_{Nx}\) to zero and solving for \(\theta\), we have: \(v_S \sin(\theta) - 1 = 0\) \(\sin(\theta) = \frac{1}{v_S} = \frac{1}{2.8}\) \(\theta = \arcsin\left(\frac{1}{2.8}\right)\)

Step 6: Calculate the value of \(\theta\)

Calculate the value of \(\theta\) using a calculator: \(\theta = \arcsin\left(\frac{1}{2.8}\right) \approx 21.8^\circ\) The swimmer should swim at an angle of \(\approx 21.8^\circ\) relative to the south bank to move directly towards the opposite shore in the north direction.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In physics, especially in problems involving vectors, trigonometry helps us break down and analyze the components of force, velocity, and more. When thinking about vectors, it's important to realize that they have both magnitude and direction. Trigonometry allows us to decompose a vector into its components using sine, cosine, and tangent functions. For instance, if you have a vector with a magnitude in a specific direction, you can find how much of that vector points along each axis.In the swimmer problem, we used trigonometric functions to express the swimmer’s velocity components. Here,

• The sine function (\(\sin(\theta)\)) was used to determine the east-west component.
• The cosine function (\(\cos(\theta)\)) helped us determine the north-south component.

This method of using trigonometry to express vector components is crucial in vector analysis, as it allows us to isolate and work with individual effects in direction.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular frame of reference. In our swimmer scenario, the relative velocity considers the speed of the swimmer with respect to the water. Why is this concept so important?

• In scenarios like the one in the exercise, the swimmer’s speed is influenced by water currents, hence the observed velocity relative to the shore (the frame of reference) is different from the swimmer's speed in stationary water.
• The concept of relative velocity helps us determine how the swimmer should compensate for the river's movement to reach the desired destination directly north.

To solve the problem, we formed a net velocity vector by adding the swimmer's velocity vector and the current's velocity. By balancing these vectors such that the east-west component becomes zero, we ensure the swimmer goes directly north, despite the east-to-west flow of the current.

Angle Calculation

Finding the correct angle is crucial for the swimmer to head directly north. This involves understanding the interaction between the different velocities. Since the river flows from east to west, the swimmer needs to adjust her angle to counteract this current.We are asked to calculate this angle (\( \theta \)), which guides her to maintain a course directly north.

• By setting the east-west component of the net velocity to zero (\( v_{Nx} = 0 \)), we derive the equation (\(v_S \sin(\theta) - 1 = 0 \)). This equation expresses what her angle must achieve: equalizing her eastward motion with the westward current.
• Solving for (\( \theta \)) using the inverse sine function gives us the specific angle needed relative to the south bank.

The calculation (\( \theta = \arcsin\left( \frac{1}{2.8} \right) \)) results in approximately 21.8°, which is the angle she should aim her swim to succeed in traveling directly north.